

































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With 

Isothermal Wall and Temperature-Dependent Viscosity 

*1Obi, Boniface Inalu 2Ohaegbulem, Emmanuel Uchenna 

 
1Department of Mathematics, Faculty of Physical Sciences, Imo State University, Owerri, 

Nigeria. 
2Department of Statistics, Faculty of Physical Sciences, Imo State University, Owerri, Nigeria. 

 

Corresponding author *1Obi, Boniface Inalu 

Abstract: 

In this research, analytical study of incompressible MHD non-Newtonian fluid in cylindrical pipe 
with isothermal wall and temperature-dependent viscosity is examined. The coupled nonlinear 
momentum and energy equations were solved using the traditional regular perturbation 
technique. Vogel’s model viscosity is introduced to account for the temperature-dependent 
viscosity, while the third grade fluid is accommodated to model the non-Newtonian fluid 
feature. It is observed that the third grade and the magnetic field parameters reduces the 
velocity profiles and increases the temperature profiles when increased at a steady rate within 
the constant viscosity index but increases the velocity and the temperature profiles when 
subjected to the Vogel model. Meanwhile the Eckert parameter is observed to enhance the 
temperature near the walls of cylindrical pipe. 
Keywords: Viscosity, Non-Newtonian, MHD, Isothermal, incompressible,  

1.0 Introduction: 

Flow of an incompressible MHD non-Newtonian fluid in cylindrical pipe finds application in 

polymer industry, petroleum industries and other types of pulp industries.In recent years, the 

non-Newtonian fluids have become very much important. However with its complexity, it is 

difficult to suggest a single model which will exhibit all the properties of non-Newtonian fluids, 

as such various empirical and semi empirical models have been put forward. Meanwhile, for 

lubricating fluids, heat generated by internal friction and the corresponding rise in temperature 

affects the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. 

Non-Newtonian fluid can be classified mainly into two groups such as differential type fluids 

and rate type fluids. Many researchers have done some work in this area amongst whom are  

Fosdick and Rajagopal [5], who examined the thermodynamics and stability of fluids of third 

grade. They showed restrictions on the stress constitutive equation. They were  concerned with 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

the relation between thermodynamics and stability for a class of non-Newtonian incompressible 

fluids of the differential type. They gave detailed attention to the special case of fluids of grade 3 

and arrived at fundamental inequalities which restricts its temperature dependent. They 

discovered that these inequalities requires that a body of such  fluid be stable in the sense that its 

total kinetic energy must tend to zero in time, no matter what its previous mechanical and 

thermal fields, provided it is both mechanically isolated and immersed in a thermally passive 

environment at constant temperature from some finite time onward. Massoudi and Christie [7] 

dealt with the effcts of variable viscosity and viscous dissipation on the flow of third grade fluid. 

The boundary layer equations of third grade fluid was treated by Pakdemirli [14]. 

 Bejan [4] studied entropy generation in fundamentally convective heat transfer. Johnson etal 

[6] investigated a fluid flow which was infused  with solid particles in a pipe, while approximate 

analytical solutions for flow of third grade  fluid was examined by Yurusoy and Pakdemirli [15]. 

Okedayo etal [12] studied the effects of viscous dissipation, constant wall temperature and a 

periodic field on unsteady flow through a horrizontal channel. Okedayo etal [13] analyzed the 

magnetohydrdynamic (MHD) flow and heat transfer in cylindrical pipe filled with porous media. 

They applied the Galerkin weighted residual method for the solution of momentum equation 

and semi- implicit finite differece method for the energy equation. They found that an increase 

in Darcy number leads to an increase in the velocity profiles, while increase in Brinkman 

number enhances the temperature of the system.Nargis and Mahmood [8] studied the 

influence of slip condition on the thin film flow of third order fluid. 

Obi [9] on approximate analytical solution of natural convection flow of non-Newtonian fluid 

through parallel plates , solved the coupled momentum and energy equations using the regular 

perturbation methd. He treated cases of constant and temperature-dependent viscosities in 

which Reynold’s and Vogel’s models were considered to account for the temperature-

dependent viscosity case, while third grade fluid was introduced to account for the non-

Newttonian effects. Obi [10] numerically analyzed the reactive third grade fluid in cylindrical 

pipe. He observed that the non-Newtonian parameters considered in the analysis: third grade 

parameter ( ), magnetic field parameter (M ), Eckert number ( Ec ) and the Brinkman number 

( Br ) had psitive effects on the velocity and temperature profiles.Aksoy and Pakdemirli [1] 

examined the flow of a non-Newtonian fluid through a porous medium between two parallel 

plates. They involed Reynold’s  and Vogel’s models viscosity and derived the criteria for validity 

for the approximate solution. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

Obi etal [11] on  semi-analytical solution of natural convection flow of non-newtonian fluid  

with temperature-dependent viscosity in pipe. They solved the nonlinear momentum and 

energy equations  using perturbation technique. They analyzed various thermo-solutal 

parameters involved in the dimensionless equations. Results within the constant viscosity show 

that increase in these parameters increases the velocity of the fluid flow as well as the 

temperature of the cylindrical pipe. It is observed that increase in the Reynold’s viscosity 

indices increases the temperature of the cylindrical pipe greatly. 

 

 

2.0 Mathematical Formulation 

Considering Aiyesimi etal [2], the steady flow of an incompressible MHD third grade fluid flow 
in a cylindrical pipe and neglecting the reacting viscous fluid assumption, the governing 
momentum and energy equations with the necessary boundary conditions can be represented 
by 

 
3

2
0                                                                    1

d du d du dp
r r B u

r dr dr r dr dr dz

 


    
            

 

   
2 2

2
3 0 0                                                         2

k d dT du du
r B u

r dr dr dr dr
  

      
                

 

         0 0 0, 0, 0                                                                              3
du dT

u a T a
dr dr

     

 

Where u is the velocity of the fluid, T is the temperature of the cylindrical pipe, T0 is the  

Plate temperature, B0 is the magnetic field,   is the coefficient of dynamic viscosity, P is 

The pressure and   is  the material coefficient relating to third grade fluid. 

The following non-dimensional variables are introduced for non-dimensionalization. 

 

 
0 0 0

, , , ,                                                                                              4
r T u

r u
d T u


 


     

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

 

Substituting equation (4) into equations (1) to (3), yields 

 
3

1
1                                                                              5

d du d du
r r Mu

r dr dr r dr dr

   
      

   
 

 
2 4

21
0                                                                6c r

d d du du
r E B Mu

r dr dr dr dr




     
        

     
 

         0 0 0, 0 0, 1 0                                                                                7
du d

u
dr dr


     

 

3.0 Method of Solution 

The semi-analytical solutions for velocity and temperature profiles can be of the form: 

                 2 2
0 1 0 1, , M= M                         8u r u r u r r r r           

 
3.1. Constant Viscosity 

 
Substituting eqn (8) into eqns (5) and (6) and separating each order of  , yields 

 0 01
: 1̀                                                                                                         9

dud
r

r dr dr


 
  

 
 

 
3

2 01
0

1
: 0  ̀                                                                           10

dudud
r r Mu

r dr dr dr


  
     

   
 

 
2

0 0 01
: 0  ̀                                                                                   11c

d dud
r E

r dr dr dr




   
    

   
 

 
4

20 01 1
0

1
: 2 2 0  ̀                                             12c r c

du dud dud
r E B E Mu

r dr dr dr dr dr




  
      

   
 

Solving eqns (9)-(12) with the condition (7), we have  

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

   

 

2 2 2 4

4 4 4 6 6

2 4

1 1 1 1 1 1 1
                                    13

4 4 392 16 48 392 24

1 1 1 1 1 1

16 16 4536 128 432 128

1 1 1 19

16 64 4536 3456

c c r

c c

u r r r M r r M

r E r E r M r r r B

E M r r E



 

  
        

  

     
           

    

 
     

 
 

1 3
                            14

288 64
r cM B E M

  
   

  

 

 

3.2.  Vogel’s Model Viscosity 

In this section, we use Vogel’s model to represent temperature – dependent viscosity and we 
apply the Massaudi and Chritie (1995) approach. The equations for momentum and energy for 
this model are: 

 
3

1
1                                                               15

d du d du d du
r r Mu

dr dr r dr dr r dr dr

 


   
      

   
 

 
2 4

21
2 0                                                          16c r

d d du du
r E B Mu

r dr dr dr dr


 

     
        

       

 exp                                                                                                       17w

Q

A
  



 
  

 
 

By Taylor series expansion of (17), we have 

 2
1                                                                                                                  18
Q

A


 

 
  

 
 

Where 

 exp                                                                                                            19w

Q

A
  

 
  

 
 

 and AQ  being parameters relating to Vogel’s model 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

 2
,                                                                                                     20

d Q d
Q q

dr A dr

 
    

Substituting eqns (8), (18) and (20) into eqns (15) and (16), we have  

 0 01
: 1̀                                                                                                    21

dud
r

r dr dr
 

 
  

 
 

 
3

30 0 0 01
02 2

1 1
: 0  ̀          22

du d du dudud q q d
r r r Mu

r dr dr A dr A dr dr r dr dr


   

      
                 

 

 
2

0 0 01
: 0  ̀                                                                                23c

d dud
r E

r dr dr dr


 

   
    

   
 

 
4

20 0 0 01 1
02

1 1
: 2 0                 24c r

q d du dud dud d
r r E B Mu

r dr dr r dr A dr dr dr dr

 
  

    
         

     
 

Solving the second order nonlinear ordinary differential eqns (21-24) with the condition (7) 
yields 

 

 

2 2 6 6

2 4 4 2 4

3 2 4

4 2 3 2 4 4 2 3

1 1 1 1 1

4 4 256 768 1152

1 1 1 1 1 1 1
         25

24 16 64 576 24 16 64

c c c

c

q q
u r r r E r E r E

A A

q
r M r r E M

A


    

      

    
         

   

     
           

     

 

   
2

4 4 8

3 2 6 6

2
4 8 3 2 4

2 6 6 2 2 2

2

6 2 2 2

1 1 1
1

64 4096 8192

1 1 1 1 1

4096 16384 9 16 64

2051 1 3
                                             

33570816 9 64

c
c

c
r

c
r

qE
r E r r r

A

qE
r r r B M r r

A

qE
B M

A

 
  

    

  

  
      

 

   
        

   


   


                              26

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

 

 

 

 

4.0 Results and Discussion 

In order to study the behaviour of some physical parameters involved in the analysis, graphs 
are presented in figures (1-10). The solution of momentum and energy equations (1) and (2) 
with the boundary condion (3) given in equations (13) and (14). Figures 1 shows the effects of 
third grade parameter on the velocity profiles. Results indicate that increase in third grade 
parameter decreases the flow velocity. This is because the third grade parameter introduces a 
relationship between the velocity and the radial distance from the boundary which is nonlinear 
and leads to drop in flow velocity. Figure 2 is the velocity profiles for various values of the 
magnetic field. It is observed from the results that increase in the magnetic field decreases the 
velocity because the applied force set in by the magnetic field parameter is perpendicular to 
the flow direction. Figure 3shows temperature profiles for different values of third grade 
parameter. Results show that as the third grade parameter increases, the temperature of the 
cylindricxal pipe increases as well owning to the thermal conductivity of the fluid which 
influences the temperature distribution. It is observed in figures 4 and 5 that the magnetic field 
and Eckert parameters increases the temperature when the parameters are increased at a 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

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“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

regular rate.Reasults further show that the Eckert parameter regulates the rate of heat 
transfer. In figures 6 and 7 show the velocity and temperature profiles for different values of 
the the third grade parameter within the Vogel model analysis respectively. Results show that 
as the third grade parameter increases, both the velocity and temperature increases at the 
same rate. In figure 8, the temperature profiles for various values of the the magnetic field 
parameter is shown. It is seen that increase in the magnetic field parameter increases the 
temperature at the walls of the cylinder. Figure 9 shows the temperature profiles for different 
values of the Vogel parameter  . It is observed that the parameter has the propensity to lower 
the temperature when increased at very small rate. Figure 10 shows the second major 
parameter A  in the Vogel model structure. Results indicate that increase in the parameter A , 
increases the temperature of the cylindrical walls. 

5.0 Conclusions 

Analytical study of incompressible MHD non-Newtonian fluid in cylindrical pipe with isothermal 

wall and temperature-dependent viscosity is examined. Vogel’s model viscosity is introduced to 

account for the temperature-dependent viscosity, while the third grade fluid is accommodated 

to model the non-Newtonian fluid feature. It is observed that the third grade and the magnetic 

field parameters reduces the velocity profiles and increases the temperature profiles when 

increased at a steady rate within the constant viscosity index but increases the velocity and the 

temperature profiles when subjected to the Vogel model. Meanwhile the Eckert parameter is 

observed to enhance the temperature near the walls of cylindrical pipe. Results further show 

that increasing  the two Vogel model indices  and A decreases and increases the temperature 

profiles respectively. 

Declarations  

1. Funding: Not applicable 

2. Informed Consent Statement: Not applicable 

3. Data Availability:  Not applicable 

4. Conflict of Interest Statement: No conflict of interest 

 

6.0  Reference 

[1] Aksoy, Y. and Pakdemirli, M.: Approximate analytical solution for flow of a third grade fluid 
through a parallel plate channel filled with a porous medium. Transp. Porous. Med. 83,375-
395(2010). 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Obi, Boniface Inalu *  

https://ijojournals.com/                                                           Volume 07 || Issue 09 || September, 2024 || 

“Analytical Study of Incompressible MHD Non-Newtonian Fluid In Cylindrical Pipe With Isothermal Wall and Temperature-Dependent Viscosity" 

 

 

[2] Aiyesimi, Y.M., Okedayo, G.T., and Obi, B.I. (2015). Flow of an incompressible MHD third 
grade 
fluid through cylindrical pipe with isothermal wall and Joule heating: Nigerian journal of 
mathematics and application 
[3] Ayub, M., Rasheed, A. and Hayat, T.,: Exact flow of third grade fluid past a porous plate 
using homotopy analysis method. International Journal of Engineering and Sciences Vol 
41,2091(2003) 
[4] Bejan, A. A study of entropy generation in fundamental convective heat transfer,ASME J. 
Heat Transfer 101, 718 (1979) 
[5] Fosdick R.L. and Rajagopal, K.R.: Thermodynamics and stability of fluids of third grade.    
Proc. R. Soc. Lond. 339, 351-377, (1980). 
[6] Johnson, G., Massoudi, M., Rajagopal, K.R.: Flow of a fluid infused with solid particles 
through a pipe. International Journal of Engineering Sciences 29, 649-661 (1991). 
[7] Massoudi, M. and Christie, I.: Effects of variable viscosity and viscous dissipation on the 
flow of a third –grade fluid in a pipe.    Int.  J. of Nonlinear Mech.,30(5): 687-699,(1995). 
[8] Nargis, K. and Mamood,T.,:The influence of slip condition on the thin film flow of a third 
grade fluid. International Journal of Nonlinear Science,30(5), 687-699(2012). 
[9] Obi B.I.: Approximate analytical study of natural convection flow of non-Newtonian fluid 
through parallel plates with heat generation. Journal of Mathematical Sciences and 
Computational Mathematics Vol.4 No.4 (2023) 
[10] Obi, B.I. Computational analysis of reactive third grade fluid in cylindrical pipe using the 
collocation method. Journal of Mathematical Sciences and Computational Mathematics. Vol.4, 
No. 4 (2023)  
[11] Obi B.I., Okorie S.I. & Nlemigwe J.C. Semi-Analytical solution of convection flow of non-

Newtonian fluid with temperature-dependent viscosity in pipe. International Journal For 

Research in Applied Science and Engineering Technology IJRASET 11(9): 782-786 (2023). 

[12] Okedayo G. T., Abah S. O and Abah R. T.: Viscous dissipation effect on the reactive flow 
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[13] Okedayo, T.G., Enenche, E. and Obi, B.I.: A computational analysis of 
magnetohydrodynamic (MHD) flow and heat transfer in cylindrical pipe filled with porous 
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Volume 07 | Issue 09 | September 2024 |            https://ijojournals.com/index.php/m/index 10


